Cambr
i
dge I
GCSE
Addi
t
i
onalMat
hemat
i
cs
Top
i
c
alQ u
e
s
t
i
on
s
St
r
ai
ghtLi
nesandGr
aphs(
Par
t2)
Cambr
i
dge I
GCSE
Addi
t
i
onalMat
hemat
i
cs
Top
i
c
alQ u
e
s
t
i
on
s
St
r
ai
ghtLi
nesandGr
aphs
(
Par
t2)
Source: 0606/21/M/J/17 - Question No. 9
1
The curve 3x 2 + xy - y 2 + 4y - 3 = 0 and the line y = 2 (1 - x) intersect at the points A and B.
(i) Find the coordinates of A and of B.
[5]
(ii) Find the equation of the perpendicular bisector of the line AB, giving your answer in the form
ax + by = c , where a, b and c are integers.
[4]
© UCLES 2017
Source: 0606/21/M/J/17 - Question No. 10
2
The table shows values of the variables t and P.
t
P
1
4.39
1.5
8.33
2
15.8
2.5
30.0
(i) Draw the graph of ln P against t on the grid below.
[2]
ln P
3
2
1
0
0.5
1
1.5
(ii) Use the graph to estimate the value of P when t = 2.2 .
2
2.5
t
[2]
(iii) Find the gradient of the graph and state the coordinates of the point where the graph meets the
vertical axis.
[2]
(iv)
Using your answers to part (iii), show that P = ab t , where a and b are constants to be found. [3]
(v)
Given that your equation in part (iv) is valid for values of t up to 10, find the smallest value of t,
correct to 1 decimal place, for which P is at least 1000.
[2]
© UCLES 2017
Source: 0606/21/M/J/18 - Question No. 8
3
An experiment was carried out recording values of y for certain values of x. The variables x and y are
thought to be connected by the relationship y = ax n , where a and n are constants.
(i) Transform the relationship y = ax n into straight line form.
[2]
The values of ln y and ln x were plotted and a line of best fit drawn. This is shown in the diagram
below.
ln y
5
4
3
2
1
0
0
2
4
6
8 ln x
(ii) Use the graph to find the value of a and of n, stating the coordinates of the points that you use. [3]
(iii) Find the value of x when y = 50.
© UCLES 2018
[2]
Source: 0606/21/M/J/19 - Question No. 10
4
Solutions to this question by accurate drawing will not be accepted.
The points A and B have coordinates ( p, 3) and (1, 4) respectively and the line L has equation 3x + y = 2 .
1
, find the value of p.
3
(i)
Given that the gradient of AB is
(ii)
Show that L is the perpendicular bisector of AB.
[3]
(iii)
Given that C `q, - 10j lies on L, find the value of q.
[1]
(iv)
Find the area of triangle ABC.
[2]
© UCLES 2019
[2]
Source: 0606/21/M/J/20 - Question No. 1
5
1
Variables x and y are such that, when 4 y is plotted against , a straight line graph passing through the
x
points (0.5, 9) and (3, 34) is obtained. Find y as a function of x.
[4]
© UCLES 2020
Source: 0606/21/O/N/18 - Question No. 10
6
The line y = 12 - 2x is a tangent to two curves. Each curve has an equation of the form
y = k + 6 + kx - x 2 , where k is a constant.
(i)
Find the two values of k.
© UCLES 2018
[5]
Source: 0606/21/O/N/18 - Question No. 10
The line y = 12 - 2x is a tangent to one curve at the point A and the other curve at the point B.
(ii)
Find the coordinates of A and of B.
[3]
(iii)
Find the equation of the perpendicular bisector of AB.
[3]
© UCLES 2018
Source: 0606/21/O/N/19 - Question No. 6
7
Do not use a calculator in this question.
The curve xy = 11x + 5 cuts the line y = x + 10 at the points A and B. The mid-point of AB is the
[7]
point C. Show that the point C lies on the line x + y = 11.
© UCLES 2019
Source: 0606/21/O/N/20 - Question No. 8
8
The population P, in millions, of a country is given by P = A # b t , where t is the number of years
after January 2000 and A and b are constants. In January 2010 the population was 40 million and had
increased to 45 million by January 2013.
(a) Show that b = 1.04 to 2 decimal places and find A to the nearest integer.
[4]
(b) Find the population in January 2020, giving your answer to the nearest million.
[1]
(c) In January of which year will the population be over 100 million for the first time?
[3]
© UCLES 2020
Source: 0606/22/M/J/16 - Question No. 5
9
The coordinates of three points are A(−2, 6), B(6, 10) and C(p, 0).
(i) Find the coordinates of M, the mid-point of AB.
[2]
(ii) Given that CM is perpendicular to AB, find the value of the constant p.
[2]
(iii) Find angle MCB.
[3]
© UCLES 2016
Source: 0606/22/M/J/17 - Question No. 8
10
Solutions to this question by accurate drawing will not be accepted.
The points A and B are (−8, 8) and (4, 0) respectively.
(i) Find the equation of the line AB.
[2]
(ii) Calculate the length of AB.
[2]
The point C is (0, 7) and D is the mid-point of AB.
(iii) Show that angle ADC is a right angle.
© UCLES 2017
[3]
Source: 0606/22/M/J/20 - Question No. 5
11
Solutions to this question by accurate drawing will not be accepted.
The points A and B are (4, 3) and (12, −7) respectively.
(a) Find the equation of the line L, the perpendicular bisector of the line AB.
[4]
(b) The line parallel to AB which passes through the point (5, 12) intersects L at the point C. Find the
coordinates of C.
[4]
© UCLES 2020
Source: 0606/22/O/N/19 - Question No. 9
12
P (r,s)
A (-3,5)
M
B (5,-1)
The diagram shows the points A (–3, 5) and B (5, –1). The mid-point of AB is M and the line PM is
perpendicular to AB. The point P has coordinates (r, s).
(i) Find the equation of the line PM in the form y = mx + c , where m and c are exact constants.
[5]
(ii) Hence find an expression for s in terms of r.
[1]
© UCLES 2019
Source: 0606/22/O/N/19 - Question No. 9
(iii) Given that the length of PM is 10 units, find the value of r and of s.
© UCLES 2019
[5]
Source: 0606/22/O/N/20 - Question No. 3
13
(a) Find the equation of the perpendicular bisector of the line joining the points (12, 1) and (4, 3),
giving your answer in the form y = mx + c .
[5]
(b) The perpendicular bisector cuts the axes at points A and B. Find the length of AB.
© UCLES 2020
[3]
Source: 0606/23/M/J/17 - Question No. 3
14
3
y
(1, 13)
(0.2, 5)
O
1
x
1
, a straight line graph passing through the
x
points (0.2, 5) and (1, 13) is obtained. Express y in terms of x.
[4]
Variables x and y are such that when 3 y is plotted against
© UCLES 2017
Source: 0606/23/M/J/17 - Question No. 3
14
3
y
(1, 13)
(0.2, 5)
O
1
x
1
, a straight line graph passing through the
x
points (0.2, 5) and (1, 13) is obtained. Express y in terms of x.
[4]
Variables x and y are such that when 3 y is plotted against
© UCLES 2017
Source: 0606/23/M/J/19 - Question No. 3
16
The points A, B and C have coordinates (4, 7), (-3, 9) and (6, 4) respectively.
(i)
Find the equation of the line, L, that is parallel to the line AB and passes through C. Give your answer
in the form ax + by = c, where a, b and c are integers.
[3]
(ii)
The line L meets the x-axis at the point D and the y-axis at the point E. Find the length of DE.
© UCLES 2019
[2]
Source: 0606/23/M/J/19 - Question No. 12
17
Do not use a calculator in this question.
The line y = 4x - 6 intersects the curve y = 10x3 - 19x2 - x at the points A, B, and C. Given that C is
the point (2, 2), find the coordinates of the midpoint of AB.
[10]
© UCLES 2019
Source: 0606/23/M/J/20 - Question No. 1
18
Solutions to this question by accurate drawing will not be accepted.
Find the equation of the perpendicular bisector of the line joining the points (4, - 7 ) and ( - 8 , 9).
© UCLES 2020
[4]
Source: 0606/23/M/J/20 - Question No. 7
19 Variables x and y are connected by the relationship y = Ax n, where A and n are constants.
(a) Transform the relationship y = Ax n to straight line form.
[2]
When ln y is plotted against ln x a straight line graph passing through the points (0, 0.5) and (3.2, 1.7)
is obtained.
(b) Find the value of n and of A.
[4]
(c)
[2]
Find the value of y when x = 11.
© UCLES 2020
Source: 0606/23/O/N/18 - Question No. 8
20
Variables x and y are such that when y2 is plotted against e2x a straight line is obtained which passes through
the points (1.5, 5.5) and (3.7, 12.1). Find
(i)
y in terms of e2x,
[3]
(ii)
the value of y when x = 3,
[1]
(iii)
the value of x when y = 50 .
[3]
© UCLES 2018
Marking Scheme
Page : 24
Question 1
1 (i)
(ii)
Question 2
2 (i)
(ii)
(iii)
(iv)
(v)
Question 3
3 (i)
(ii)
(iii)
Question 4
4 (i)
(ii)
(iii)
(iv)
Question 5
5
Question 6
6 (i)
(ii)
(iii)
Question 7
7
Question 8
8 (a)
(b)
(c)
Question 9
9 (i)
(ii)
(iii)
Question 10
10 (i)
(ii)
(iii)
Question 11
11 (a)
(b)
Question 12
12 (i)
(ii)
(iii)
Question 13
13 (a)
(b)
Question 14
14
Question 15
15 (i)
(ii)
(iii)
Question 16
16 (i)
(ii)
Question 17
17
Question 18
18
Question 19
19 (a)
(b)
(c)
Question 20
20 (i)
(ii)
(iii)